Isoperimetric Theorem for Quadrilaterals

Introduction

Among all quadrilaterals with the same perimeter, square has the largest area.

or, equivalently,

Among all quadrilaterals with the same area, square has the smallest perimeter.

The proof is in three steps:

  1. For any quadrilateral (that is not a parallelogram), there is a parallelogram with smaller perimeter but the same area.

  2. For every parallelogram (that is not a rectangle), there is a rectangle with larger area but the same perimeter.

  3. For every rectangle (that is not a square) there is a square of larger area but the same perimeter.

Parallelogram among Quadrilaterals

For any quadrilateral (that is not a parallelogram), there is a parallelogram with smaller perimeter but the same area.

Among all quadrilaterals with the same perimeter parallelogram encloses the largest area

Indeed.

Rectangle among Parallelograms

For every parallelogram, there is a rectangle with larger area but the same perimeter.

among all parallelograms with the same perimeter rectangle encloses the largest area

Indeed.

Square among Rectangles

For every rectangle (that is not a square) there is a square of larger area but the same perimeter.

Among all rectangles with the same perimeter square has the largest area

Indeed.

Acknowledgment

The above has been communicated to me in private correspondence by Sidney Kung on 1 March, 2014.

Related material
Read more...

  • Isoperimetric Theorem and Inequality
  • An Isoperimetric theorem
  • Isoperimetric theorem and its variants
  • Isoperimetric Property of Equilateral Triangles
  • Isoperimetric Property of Equilateral Triangles II
  • Maximum Area Property of Equilateral Triangles
  • Isoperimetric Theorem For Quadrilaterals
  • Parallelograms among Quadrilaterals
  • Rectangles among Parallelograms
  • Isoperimetric Theorem for Rectangles
  • Isoperimetric Theorem For Quadrilaterals II
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